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Hollow high-quality electron beam generation by cyclotron autoresonance in laser-induced acceleration
Optik - International Journal for Light and Electron Optics xxx (xxxx) xxxx
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Original research article
Hollow high-quality electron beam generation by cyclotron autoresonance in laser-induced acceleration S.H. Hashemian, H. Akou* Department of Physics, Faculty of Basic Science, Babol Noshirvani University of Technology, Babol 47148-71167, Iran
ABS TRA CT
In this study, the dynamics of electrons by a focused Gaussian laser pulse in the presence of an ultrashort axial magnetic field have been investigated numerically. The cyclotron resonance of the electrons due to the external magnetic field were theoretically investigated. An optimized magnetic field can enhance the electron energy gradient with ultra-low scattering. The simulation provides a circulating hollow electron beam suitable for high-precision and high-energy particle physics experiments. The features of output electron beam such as mean energy gain, energy spectra, spatial emittance and angular scattering distribution were examined and their concerning physical explanations have been presented. It has been found that an electron bunch with about MeV initial energy becomes a compressed electron beam with about 3 GeV energy, after interacting with the laser pulse applied by a nanosecond pulsed magnetic field.
1. Introduction In recent years, due to advances in high-power laser technology [1], many studies have been done to develop charged particle laser accelerators as an alternative for traditional accelerators. A great deal of research has explored the techniques for higher energy gains through laser-driven electron acceleration in vacuum [2–6]. To achieve higher energy and lower scattering, in direct laser acceleration (DLA), some approaches such as using chirped laser pulse [7–17] and cyclotron auto-resonance by applying magnetic fields [18–27] have also been introduced. For Example: Gupta and Suk [21] discussed the combined effect of an axial static magnetic field and a frequency chirp on single electron energy gain. They predicted that an electron with 50 MeV initial kinetic energy can gain an energy of about 100 GeV in interaction with a laser pulse of intensity a 0 = 100 . Singh [22] investigated the dynamics of a single electron interacting with a circularly polarized laser pulse influenced by a short duration magnetic field. He obtained that the value of optimum magnetic field is independent of the laser intensity and decreases with initial electron energy. Akou and Hamedi [25] studied the combined effect of chirping laser frequency and an axial static magnetic field on the electron bunch acceleration. It is shown that a micro electron bunch with energy of the order of GeV can be generated by using a laser pulse with intensity of 6.7 × 1019 W cm −2 influenced by a magnetic field with strength of 100 T. In this work, we apply an external time-dependent magnetic field with Gaussian temporal profile and present the configuration of a single electron interaction with a femto-second high intensity laser pulse in detail. Then, by considering a pre-accelerated electron bunch with 1.17 MeV initial mean energy injected to an appropriate position in front of the laser pulse, many particle simulation is studied. The simulation demonstrates that a high quality electron beam in multi-GeV mean energy and very low spatial scattering will be produced. A circulating hollow electron beam is generated by the simulation, which is very important, because it may be used in collimation in storage rings and colliders [29]. Such circulating electron beam with increasing longitudinal angular momentum are also expected to provide new capacities for electron microscopy [30] and other applications [31–34]. This paper is organized as follows. In Section II, the governing equations for the laser electron acceleration in vacuum are given.
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Corresponding author. E-mail address: [email protected] (H. Akou).
https://doi.org/10.1016/j.ijleo.2019.163708 Received 24 July 2019; Received in revised form 21 October 2019; Accepted 5 November 2019 0030-4026/ © 2019 Elsevier GmbH. All rights reserved.
Please cite this article as: S.H. Hashemian and H. Akou, Optik - International Journal for Light and Electron Optics, https://doi.org/10.1016/j.ijleo.2019.163708
Optik - International Journal for Light and Electron Optics xxx (xxxx) xxxx
S.H. Hashemian and H. Akou
Simulation results and discussion are presented in Section III. Finally, summary and conclusions are given in Section IV. 2. Governing equations In our simulation model, the lowest-order Hermite–Gaussian mode with spatial and temporal Gaussian profiles is considered. It is assumed that the laser pulse propagates in z -direction and is right-handed circularly polarized (RHCP) in the transverse plane. The transverse components of the laser electric field, in the paraxial approximation, are given by
Ex =
E0 r2 exp ⎡− 2 2 + iΦ(r , z , t ) ⎤ g (ct − z ), ⎢ r0 f (z ) ⎥ f (z ) ⎣ ⎦
(1)
is the and Ey = Ex exp[−iπ /2], where E0 is the amplitude of laser electric field, r0 is the minimum spot size of the laser, z r = f (z ) = 1 + (z / z r )2 r 2 = x 2 + y2, is the Gaussian laser beam width parameter, and Rayleigh length, Φ(r , z , t ) = ϕr (r , z ) + ϕG (z ) + ϕp (z , t ) + ϕ0 is the total phase of the Gaussian beam including the radially phase ϕr (r , z ) = zr 2/ z r r02 f 2 , the Gouy phase shift ϕG (z ) = −tan−1 (z / z r ) , the plane wave phase ϕp (z , t ) = k 0 z − ω0 t and initial phase ϕ0 . For focused Gaussian laser beam there exists a longitudinal component for the electric field due to satisfying Maxwell's equation, ∇ . EL = 0 [35,36], which can be obtain by Ez = (i/ k 0)(∂Ex / ∂x + ∂Ey / ∂y ) , where k 0 is laser wave number. In Eq. (1) g (ct − z ) represents the temporal envelope of the laser pulse which is assumed Gaussian that given by
k 0 r02/2
2
⎡ ct − (z − zL) ⎞ ⎤ g (ct − z ) = exp ⎢−⎛⎜ ⎟ ⎥, cτp ⎠⎦ ⎣ ⎝
(2)
where, c is the speed of light, zL is the initial position of the pulse peak and τp is the laser pulse duration. The magnetic field components related to the laser pulse are given by one another Maxwell's equations, ∂B L / ∂t = −c∇ × EL . Apart form the laser electric and magnetic fields, an externally pulsed magnetic field is also applied in the longitudinal direction which is given by 2
⎛ t − t0 ⎞ Bext (t ) = k B0 exp ⎡ ⎢− τb ⎠ ⎣ ⎝
ˆ
⎜
⎟
⎤, ⎥ ⎦
(3)
where, B0 is the strength of the external magnetic field, τb and t0 represent the time duration and the peak position of the magnetic field pulse, respectively. From Maxwell's equations, the electric field related to the time-dependent pulsed magnetic field can be derived which we neglect it in this simulation because it is very small as compared to laser electric field [28]. The dynamics of an electron interacting with the laser pulse and the external magnetic field are given by the relativistic Newton–Lorentz equation of motion
dP v = −e EL + × [B L + Bext (t )] , dt c
{
}
(4)
and the equation governing to the electron energy gain
d (γm 0e c 2) = −e EL . v , dt
(5)
v 2/ c 2
where P = γm 0e v is the electron's relativistic momentum, me0 is the electron rest mass, γ = 1/ 1 − is the relativistic factor and v is the electron's velocity. By decomposing of Eqs. (4) and (5), the equations of particle motion are complete with the equations governing to the electron's trajectory,
dr 1 = P. dt γm 0e
(6)
Eqs. (4)–(6) are a set of coupled ordinary differential equations which have been solved numerically by a 3D test particle code in FORTRAN using the fourth-order Runge–Kutta method. For numerically simulation we normalize the spatial and temporal quantities with k 0 and ω0 , respectively, velocity with c and electric and magnetic field with e / m 0e cω0 . 3. Simulation results and discussion It is assumed that a pre-accelerated electron is initially injected at an angle δ with respect to the propagation direction of the laser pulse. Hence, the electron velocity is given by v0 = ˆi v0 sin δ + kˆ v0 cos δ , where v0 = 0.9c is the electron's initial velocity. It is also assumed that interaction begins when the electron reaches the origin ( x 0 = y0 = z 0 = 0 ) while the peak of the laser pulse is located at zL = −12.73 μ m. The evolution of the electron's trajectory and velocity can be obtained by simultaneous solution of Eqs. (4)–(6). The parameters of the laser pulse and the external magnetic field used in the simulation, have been presented in Table 1 and the rest parameters can be optimized by exploring dynamics of the single electron. For search of parameters that may lead to higher energy gains, in Fig. 1 we have examined the variation of the electron energy gain as a function of different parameters related to the pulsed magnetic field (according to Eq. (3)). Fig. 1a shows the variation of the electron energy gain as a function of B0 for fixed parameters t0 = 0 and τb = 424 ps. It clearly shows there are two peaks for electron energy curve and the maximum energy gain, about ε = 4.5 2
Optik - International Journal for Light and Electron Optics xxx (xxxx) xxxx
S.H. Hashemian and H. Akou
Table 1 Parameters of the laser pulse and the external magnetic field. Pulsed magnetic field
Gaussian laser pulse
Strength; B0 = 0.415 (5.6 kT) Peak of the pulse; t0 = 0
Wave length; λ 0 = 0.8 μ m Time duration; τp = 22 fs Beam waist; r0 = 40 μ m
Life time; τ b = 0.424 ns
Laser intensity; I = 2.5 × 1020 W cm −2 Initial phase; ϕ0 = 0
Fig. 1. Final electron energy gain as a function of a) the strength of external magnetic field B0 , b) the peak of the magnetic field pulse t0 , c) the life time of magnetic filed τb .
GeV, can be achieved when the normalized magnetic field B0 is set to 0.37. The relationship between ε and t0 for fixed parameters B0 = 0.415 and τb = 424 ps was also investigated, which is given in Fig. 1.b. As we indicated, t0 represents the peak position of the magnetic field pulse in time. As can be seen, the final electron energy is maximized at t0 = 148 ps and after the point, the electron energy gain is decreased by increasing t0 . In Fig. 1c, t0 = 0 and B0 = 0.415[=5.6 kT] were held constant and the duration of the pulsed external magnetic field is varied. Such value of the high intensity magnetic field with short life time (nanosecond) is currently available experimentally [37,38]. As shown in Fig. 1c, the electron energy increases exponentially with increasing the magnetic field pulse duration. This was predictable, because the increase in the duration of the pulsed magnetic field would mean increasing the time that the magnetic field affects the particle and consequently, the particle remains in the acceleration phase. By optimization, it is seen that an electron with 10.5 GeV energy can be obtained by a RHCP laser pulse with intensity 3.8 × 1021W/cm 2 corresponding to a0 = 0.61 × 10−9λ 0 [μm] I [W /cm2] = 30 in the presence of the pulsed magnetic field with B0 = 0.6, t0 = 0 and τb = 424 ps. By considering the Doppler shift effect, and by applying the external magnetic field the cyclotron resonance condition between the electron and the laser wave will be ω0 − k 0 vz = eBext (t )/ γm 0e c ≡ ωc (t )/ γ , where ωc (t ) is the non-relativistic cyclotron frequency [24]. Using dimensionless variables the resonance condition can be written as
γ (1 − βz ) Bext (t )
= 1,
(7)
Fig. 2. a) Variation of normalized external magnetic field Bext (t ) (red line) and electron energy gain ε (blue line) versus the interaction time. b) Variation of the normalized external magnetic field Bext (t ) (red line) and γ (1 − βz ) which presents the numerator of Eq. (21) (blue line) versus the interaction time. The inset shows the variation of γ (1 − βz )/ Bext (t ) versus time. c) Evolution of energy gradient dε / dt versus time. 3
Optik - International Journal for Light and Electron Optics xxx (xxxx) xxxx
S.H. Hashemian and H. Akou
where Bext (t ) is the normalized external magnetic field. Fig. 2.a shows the pulsed external magnetic field in red line and the variation of electron energy gain in blue line as a function of the interaction time t . The maximum energy gain of 4.4 GeV is observed for the electron. To check the resonance condition, the variation of numerator and denominator of Eq. (7), versus interaction time are shown in Fig. 2.b. As can be seen, these two curves cross each other at t = 56.878 ps. This is the time which resonance condition occurs exactly as seen in the inset of Fig. 2b which shows the variation of the left side of Eq. (7). For further investigation of resonance, it is useful to consider the behaviour of the electron energy gradient over the interaction time. Using Eq. (5) the energy gradient of the electron interacting with the laser wave can given by
dε e EL . v =− . dz vz
(8)
The evolution of the acceleration gradient is shown in Fig. 2c. It is seen that exactly after t = 56.878 ps from the beginning of the interaction, the electron energy gradient reaches its maximum value. Although, in the rest of the interaction time, the auto-resonance condition does not exactly take place, in the most important part of the electron acceleration process, (0 < t < 200 ps) the fraction in the left side of Eq. (7) is not very different from 1 and the acceleration gradient is positive. In fact after this stage, the interaction practically finished. It is interesting to note that the gradient of energy is much larger than that in the conventional accelerators. After the detailed investigation of the single electron dynamics, we now consider a bunch of electrons injected into the laser acceleration configuration in the presence of the pulsed magnetic field. A random distribution of an ensemble of non-interacting electrons is considered within a volume of cylindrical shape along the z -direction. The coordinates of the electrons ( x , y, z ) are defined by generating random numbers for ρ , φ and z in cylindrical coordinate system as 0 ≤ ρ ≤ RB , 0 ≤ φ ≤ 2π and 0 ≤ z ≤ LB . Where RB and LB are respectively the radius and length of the cylinder containing the electrons. In this simulation, a normal distribution has been used to generate the random numbers for ρ , while a uniform distribution has been selected for φ and z . Therefore, the electrons in the cylinder are randomly Gaussian distributed around the z -axis. To determine the components of the initial velocity of each electron in the initial beam the following relations are used.
βx = |β0 |sin δ cos φδ , ⎧ ⎪ βy = |β0 |sin δ sin φδ , ⎨ ⎪ βz = |β0 |cos δ . ⎩ where |β0 | is the initial velocity of an electron, δ and φδ represent the axial and azimuthal angles. As can be seen in Fig. 3.a, the initial velocity of the electrons follows a normal (Gaussian) distribution with mean value β¯0 = 0.9 and standard deviation Δβ0 = 0.013. Fig. 3b shows the distribution of initial energy of the electrons. The initial mean energy of the electron beam is ε¯i = 1.17 MeV with deviation Δεi = 0.06 MeV. Such the pre-accelerated electron beam can be achieved using a short LINAC or a tabletop betatron. The space distribution of initial and output electron beam have been shown in Fig. 4a and b. The electron number density of the initial bunch is n = 3.2 × 1011cm −3 and the radial and longitudinal sizes of the cylinder are RB = 10 μm = 0.25r0 and LB = 20 μm = 25λ 0 , respectively. For further clarification of the longitudinal distribution of the output electrons, the figure is magnified along the z -direction as can be seen in Fig. 4c. According to the figures, the features of the outgoing electron beam have been presented as follows. 3.1. Hollow electron beam As can be seen in the cross-section of the electrons distribution in the x − y plane (Fig. 4b and c), the accelerated electrons leave around the z -axis toward the radial direction, therefore a hollow electron beam is formed. The cross-section forms a circular ring with a radius 30 μ m and a thickness of 20 μ m. This feature is due to the cyclotron rotation of the electrons around the axial magnetic field
Fig. 3. Percentage of total number of the injected electrons as a function of a) initial velocity β0 and b) initial energy εi . 4
Optik - International Journal for Light and Electron Optics xxx (xxxx) xxxx
S.H. Hashemian and H. Akou
Fig. 4. a) Initial spatial distribution of electron bunch in a cylinder. b) 3D spatial distribution of the outgoing electrons and c) the magnified spatial distribution in z-direction.
which can cause the electrons to be expelled from the z -direction. Moreover, the radially ponderomotive force related to the Gaussian laser beam can push the electrons away from the axis. Fig. 5.a illustrates the percentage of total number of outgoing electrons versus radial coordinate r . It is seen that due to the ponderomotive force, the transverse plane in 0 < r < 10 μ m is empty of each particle and more particles are distributed around the r = 24 μ m. Such hollow electron beam may be used in collimation in storage rings and colliders [29] 3.2. Circulating electron beam Due to the rotation of each electron around the propagation direction caused by the applied external magnetic field, the ensemble of the particles performs a general rotational movement. This is clear in Fig. 4c. Therefore, a circulating electron beam has been produced which moves forward with high relativistic velocities. The evolution of angular momentum of electrons in z -direction is responsible for the circulation process in the transverse plane. The longitudinal component of the electron angular momentum is given by Lz = [r × P]z = γ [xβy − yβx ], which indicates all electrons rotate around the axis. The electron beam with carrying axial angular momentum can provide new capabilities for electron microscopy [30] and micro-manipulation [31,32]. 3.3. Transverse emittance Emittance is a fundamentally important parameter which can evaluate quality of a bunch of particles. That is a criterion for the particle scattering in transverse directions. The root-mean-square (RMS) emittance in the transverse direction can be obtained from the following formula [39] 1
1
ϵx = 〈 (x − 〈x 〉)2〉2 〈 (x ′ − 〈x ′〉)2〉2 ,
(9)
Fig. 5. a) Percentage of total number of the electrons in output beam versus radial coordinate. b) The spread distribution of outgoing electron beam energy. 5
Optik - International Journal for Light and Electron Optics xxx (xxxx) xxxx
S.H. Hashemian and H. Akou
where x ′ = Px / Pz is the inclination of the electron trajectory relative to the z -axis which denotes the divergence of electrons and Px is the electron momentum. In our calculation, the initial and final transverse RMS emittances in π mm.mrad unit are respectively ϵix , y = 0.0092 and ϵ xf , y = 0.045. Thus, the ratio of the emittances of the final electron beam to the initial one is ϵ xf , y /ϵix , y = 4.89. Such values for emittance of an electron bunch are very favourable compared to other theoretical or experimental reports. 3.4. Energy distribution Another important parameter to indicate the quality of the accelerated electron bunch is its energy distribution. The simulation results show that the outgoing electrons form a high energy bullet with a length of 4λ 0 in z -direction. The mean energy gain of the electron beam is ε¯ = 3.2 GeV. Since laser wave phases experienced by each electron may be different, their energy gain is slightly different. Therefore, the energy distribution of the output beam is broadened. The broadening of the energy distribution of the output electrons has been shown in Fig. 5.b. It is seen that the energy of more electrons is about 3.3 GeV. By calculating the FWHM of the electron number in Fig. 5.b the beam energy spread is estimated to be about (Δε )1/2 = 0.5 GeV. Now we can calculate the energy emittance of the output electrons by ϵ′ = (Δε )1/2 / εp [2], where εp is the energy corresponding to the peak of energy spectrum. The energy emittance of the output electron beam is ϵ′ = 0.15, which shows that the energy divergence between the particles is very small in comparison with the other works [2,25]. This is an important issue in practical application. 3.5. Angular spectra There is a simple correlation between the scattering angle and the escape energy of the electron. The scattering angle of an output electron can be obtain by δf = arctan(P⊥/ Pz ) . Using energy conservation and relativistic factor one can also write [40]
⎡ γ2 − 1 ⎤ δf = tan−1 ⎢ − 1 ⎥. (γβz )2 ⎣ ⎦
(10)
Fig. 6.a presents the angular distribution of the outgoing electrons. As shown in the figure, the electrons are concentrated in a smallscattering angle region. In this simulation the mean scattering angle is about δ¯f = 0.66°. In Fig. 6b, we show the scattering angle of the outgoing electrons as a function of the injection angle of the electrons δi . As can be seen, while the injection angle of the electrons is in the range of 0 to 2° (Δδi = 2°), their scattering angle range is between 0.62° to 0.82° (Δδf = 0.2°). In other words, the accelerated electrons are distributed in a very small angular range. The characteristics of the initial and outgoing electron beam have been summarized in Table 2. 4. Summary and conclusions In this paper, the interaction of a single electron with a femto-second RHCP Gaussian laser pulse in the presence of a short duration axial magnetic field has been considered. The laser peak intensity is about 2.5 × 1020 W cm −2 (peak power 6.3 PW), and the peak strength of the nano-second pulsed magnetic field is about 5.6 kT . Such employed magnetic fields are currently available in spite of their great strength because of their ultra-short lifetime (τb < 1 ns) [37,38]. Due to the magnetic field, the electron rotates around the z -axis with cyclotron frequency. It is concluded that, when the cyclotron frequency coincides with the laser frequency experienced by the relativistic electron, a resonance occurs and therefore it leads to a maximal energy gradient, and a multi-GeV energy exchange between the laser wave and electron. The main result of this paper is the development of the production of a high quality electron beam with several special and novel features. The simulation presents a high-quality circulating hollow electron beam. The
Fig. 6. a) Angular distribution of outgoing electrons. b) The scattering angle of outgoing electrons (δf ) as a function of the injection angle of the electrons (δi ). 6
Optik - International Journal for Light and Electron Optics xxx (xxxx) xxxx
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Table 2 Features of the initial and output electron beam. Initial Beam
Output Beam
1.17 MeV 0.05 0.0092
3.2 GeV 0.15 0.045
Bunch diameter dB (=2RB ) Mean injection (scattering) angle δ Injection (Scattering) angle range Δδ
20 μm (≃25λ 0) 20 μ m 0.46° 2°
3.2 μm (≃4λ 0) 60 μ m 0.66° 0.2°
Electron number density [cm−3 ]
3.2 × 1011
1.8 × 1011
Mean energy ε¯ Energy emittance ϵ′[Dimensionless] Transverse Emittance ϵx , y [π mm.mrad] Bunch length LB
quality of the beam depends on its energy gain, energy spread and the particle's scattering where the last two characteristics can be expressed by energy and spatial emittances. The simulation reveals a high-energy electron beam with a mean energy of about ε¯ = 3.2 GeV and with an low energy emittance (ϵ′ = 0.15), suitable for high-precision particle physics experiments. The presence of a suitable pulsed magnetic field during the laser-particle interaction not only improves the beam features related to energy, but also amends the spatial emittance as well as the angular scattering of the electrons. The transverse emittance of the magnetized output beam is about 0.045 π mm.mrad, which compares very small with what is obtained from other theoretical reports [23,25] and conventional accelerators [41]. The simulation results show that the broadening of the electrons scattering angle relative to the broadening of the electrons injection angle is Δδf /Δδi = 0.1 which means that the outgoing electrons are distributed in a smaller angular range compared to the injection electrons. As a final word, the shape of output electron beam and its rotation add to its the practical applications. Acknowledgements The authors acknowledge the funding support of Babol Noshirvani University of Technology through Grant program No. BNUT/ 391040/98. References [1] V. Yanovsky, V. Chvykov, G. Kalinchenko, P. Rousseau, T. Planchon, T. Matsuoka, A. Maksimchuk, J. Nees, G. Cheriaux, G. Mourou, K. Krushelnick, Opt. Exp. 16 (2008) 2109 Central Laser Facility www.clf.rl.ac.uk/.. [2] P.X. Wang, Y.K. Ho, J. Pang, X.Q. Yuan, Q. Kong, N. Cao, Nucl. Instrum. Methods Phys. Res. Sect., A 482 (2002) 581. [3] D.N. Gupta, N. Kant, D.E. Kim, H. Suk, Phys. Lett. A 368 (2007) 402. [4] Y.I. Salamin, Phys. Rev. A 73 (2006) 043402. [5] P.B. Corkum, F. Krausz, Nat. Phys. 3 (2007) 381. [6] H. Zhang, Sh. Liu, H. Guo, Phys. Lett. A 367 (2007) 402. [7] A.G. Khachatryan, F.A. van Goor, K.-J. Boller, Phys. Rev. E 70 (2004) 067601. [8] X.Y. Wu, P.X. Wang, S. Kawata, Appl. Phys. Lett. 100 (2012) 221109. [9] M. Asri, H. Akou, Optik 143 (2017) 142. [10] Y.I. Salamin, Phys. Lett. A 376 (2012) 2442. [11] M. Akhyani, F. Jahangiri, A.R. Niknam, R. Massudi, J. Appl. Phys. 118 (2015) 183106. [12] F. Sohbatzadeh, S. Mirzanejhad, H. Aku, Phys. Plasmas 16 (2009) 023106. [13] F. Sohbatzadeh, S. Mirzanejhad, H. Aku, S. Ashouri, Phys. Plasmas 17 (2010) 083108. [14] F. Sohbatzadeh, H. Aku, J. Plasma Phys. 77 (2011) 39. [15] H. Akou, Phys. Plasmas 25 (2018) 063105. [16] M. Fouladi, H. Akou, J. Opt. Soc. Am. B 36 (2019) 603. [17] H. Akou, Optik 131 (2017) 446. [18] H. Liu, X.T. He, S.G. Chen, Phys. Rev. E 69 (2004) 066409. [19] D.N. Gupta, C.M. Ryu, Phys. Plasmas 12 (2005) 053103. [20] K.P. Singh, Phys. Rev. E 69 (2004) 056410. [21] D.N. Gupta, H. Suk, Phys. Plasmas 13 (2006) 013105. [22] K.P. Singh, J. Appl. Phys. 100 (2006) 044907. [23] B.J. Galow, J.-X. Li, Y.I. Salamin, Z. Harman, Ch.H. Keitel, Phys. Rev. ST. Accel. Beams 16 (2013) 081302. [24] H. Saberi, B. Maraghechi, Phys. Plasmas 22 (2015) 033115. [25] H. Akou, M. Hamedi, Phys. Plasmas 22 (2015) 103120. [26] H.S. Ghotra, N. Kant, Phys. Plasmas 23 (2016) 013101. [27] H.S. Ghotra, N. Kant, Phys. Plasmas 23 (2016) 053115. [28] D.N. Gupta, H. Suk, Appl. Phys. Lett. 91 (2007) 211101. [29] G. Stancari, A. Valishev, G. Annala, G. Kuznetsov, V. Shiltsev, D.A. Still, L.G. Vorobiev, Phys. Rev. Lett. 107 (2011) 084802. [30] S. Furhapter, A. Jesacher, S. Bernet, M. Ritsch-Marte, Opt. Lett. 30 (2005) 1953. [31] D.G. Grier, Nature 424 (2003) 810. [32] A. Mair, A. Vaziri, G. Weihs, A. Zeilinger, Nature 412 (2001) 313. [33] G. Foo, D.M. Palacios, G.A. Swartzlander, Opt. Lett. 30 (2005) 3308. [34] M.F. Anderson, C. Ryu, P. Clad, V. Natarajan, A. Vaziri, K. Helmerson, W.D. Phillips, Phys. Rev. Lett. 97 (2006) 170406. [35] K.T. McDonald, Phys. Rev. Lett. 80 (1998) 1350. [36] P. Mora, B. Quesnel, Phys. Rev. Lett. 80 (1998) 1351. [37] K.F.F. Law, M. Bailly-Grandvaux, A. Morace, S. Sakata, K. Matsuo, S. Kojima, S. Lee, X. Vaisseau, Y. Arikawa, A. Yogo, K. Kondo, Z. Zhang, C. Bellei, J.J. Santos, S. Fujioka, H. Azechi, Appl. Phys. Lett. 108 (2016) 091104.
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Optik journal homepage: www.elsevier.com/locate/ijleo
Original research article
Hollow high-quality electron beam generation by cyclotron autoresonance in laser-induced acceleration S.H. Hashemian, H. Akou* Department of Physics, Faculty of Basic Science, Babol Noshirvani University of Technology, Babol 47148-71167, Iran
ABS TRA CT
In this study, the dynamics of electrons by a focused Gaussian laser pulse in the presence of an ultrashort axial magnetic field have been investigated numerically. The cyclotron resonance of the electrons due to the external magnetic field were theoretically investigated. An optimized magnetic field can enhance the electron energy gradient with ultra-low scattering. The simulation provides a circulating hollow electron beam suitable for high-precision and high-energy particle physics experiments. The features of output electron beam such as mean energy gain, energy spectra, spatial emittance and angular scattering distribution were examined and their concerning physical explanations have been presented. It has been found that an electron bunch with about MeV initial energy becomes a compressed electron beam with about 3 GeV energy, after interacting with the laser pulse applied by a nanosecond pulsed magnetic field.
1. Introduction In recent years, due to advances in high-power laser technology [1], many studies have been done to develop charged particle laser accelerators as an alternative for traditional accelerators. A great deal of research has explored the techniques for higher energy gains through laser-driven electron acceleration in vacuum [2–6]. To achieve higher energy and lower scattering, in direct laser acceleration (DLA), some approaches such as using chirped laser pulse [7–17] and cyclotron auto-resonance by applying magnetic fields [18–27] have also been introduced. For Example: Gupta and Suk [21] discussed the combined effect of an axial static magnetic field and a frequency chirp on single electron energy gain. They predicted that an electron with 50 MeV initial kinetic energy can gain an energy of about 100 GeV in interaction with a laser pulse of intensity a 0 = 100 . Singh [22] investigated the dynamics of a single electron interacting with a circularly polarized laser pulse influenced by a short duration magnetic field. He obtained that the value of optimum magnetic field is independent of the laser intensity and decreases with initial electron energy. Akou and Hamedi [25] studied the combined effect of chirping laser frequency and an axial static magnetic field on the electron bunch acceleration. It is shown that a micro electron bunch with energy of the order of GeV can be generated by using a laser pulse with intensity of 6.7 × 1019 W cm −2 influenced by a magnetic field with strength of 100 T. In this work, we apply an external time-dependent magnetic field with Gaussian temporal profile and present the configuration of a single electron interaction with a femto-second high intensity laser pulse in detail. Then, by considering a pre-accelerated electron bunch with 1.17 MeV initial mean energy injected to an appropriate position in front of the laser pulse, many particle simulation is studied. The simulation demonstrates that a high quality electron beam in multi-GeV mean energy and very low spatial scattering will be produced. A circulating hollow electron beam is generated by the simulation, which is very important, because it may be used in collimation in storage rings and colliders [29]. Such circulating electron beam with increasing longitudinal angular momentum are also expected to provide new capacities for electron microscopy [30] and other applications [31–34]. This paper is organized as follows. In Section II, the governing equations for the laser electron acceleration in vacuum are given.
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Corresponding author. E-mail address: [email protected] (H. Akou).
https://doi.org/10.1016/j.ijleo.2019.163708 Received 24 July 2019; Received in revised form 21 October 2019; Accepted 5 November 2019 0030-4026/ © 2019 Elsevier GmbH. All rights reserved.
Please cite this article as: S.H. Hashemian and H. Akou, Optik - International Journal for Light and Electron Optics, https://doi.org/10.1016/j.ijleo.2019.163708
Optik - International Journal for Light and Electron Optics xxx (xxxx) xxxx
S.H. Hashemian and H. Akou
Simulation results and discussion are presented in Section III. Finally, summary and conclusions are given in Section IV. 2. Governing equations In our simulation model, the lowest-order Hermite–Gaussian mode with spatial and temporal Gaussian profiles is considered. It is assumed that the laser pulse propagates in z -direction and is right-handed circularly polarized (RHCP) in the transverse plane. The transverse components of the laser electric field, in the paraxial approximation, are given by
Ex =
E0 r2 exp ⎡− 2 2 + iΦ(r , z , t ) ⎤ g (ct − z ), ⎢ r0 f (z ) ⎥ f (z ) ⎣ ⎦
(1)
is the and Ey = Ex exp[−iπ /2], where E0 is the amplitude of laser electric field, r0 is the minimum spot size of the laser, z r = f (z ) = 1 + (z / z r )2 r 2 = x 2 + y2, is the Gaussian laser beam width parameter, and Rayleigh length, Φ(r , z , t ) = ϕr (r , z ) + ϕG (z ) + ϕp (z , t ) + ϕ0 is the total phase of the Gaussian beam including the radially phase ϕr (r , z ) = zr 2/ z r r02 f 2 , the Gouy phase shift ϕG (z ) = −tan−1 (z / z r ) , the plane wave phase ϕp (z , t ) = k 0 z − ω0 t and initial phase ϕ0 . For focused Gaussian laser beam there exists a longitudinal component for the electric field due to satisfying Maxwell's equation, ∇ . EL = 0 [35,36], which can be obtain by Ez = (i/ k 0)(∂Ex / ∂x + ∂Ey / ∂y ) , where k 0 is laser wave number. In Eq. (1) g (ct − z ) represents the temporal envelope of the laser pulse which is assumed Gaussian that given by
k 0 r02/2
2
⎡ ct − (z − zL) ⎞ ⎤ g (ct − z ) = exp ⎢−⎛⎜ ⎟ ⎥, cτp ⎠⎦ ⎣ ⎝
(2)
where, c is the speed of light, zL is the initial position of the pulse peak and τp is the laser pulse duration. The magnetic field components related to the laser pulse are given by one another Maxwell's equations, ∂B L / ∂t = −c∇ × EL . Apart form the laser electric and magnetic fields, an externally pulsed magnetic field is also applied in the longitudinal direction which is given by 2
⎛ t − t0 ⎞ Bext (t ) = k B0 exp ⎡ ⎢− τb ⎠ ⎣ ⎝
ˆ
⎜
⎟
⎤, ⎥ ⎦
(3)
where, B0 is the strength of the external magnetic field, τb and t0 represent the time duration and the peak position of the magnetic field pulse, respectively. From Maxwell's equations, the electric field related to the time-dependent pulsed magnetic field can be derived which we neglect it in this simulation because it is very small as compared to laser electric field [28]. The dynamics of an electron interacting with the laser pulse and the external magnetic field are given by the relativistic Newton–Lorentz equation of motion
dP v = −e EL + × [B L + Bext (t )] , dt c
{
}
(4)
and the equation governing to the electron energy gain
d (γm 0e c 2) = −e EL . v , dt
(5)
v 2/ c 2
where P = γm 0e v is the electron's relativistic momentum, me0 is the electron rest mass, γ = 1/ 1 − is the relativistic factor and v is the electron's velocity. By decomposing of Eqs. (4) and (5), the equations of particle motion are complete with the equations governing to the electron's trajectory,
dr 1 = P. dt γm 0e
(6)
Eqs. (4)–(6) are a set of coupled ordinary differential equations which have been solved numerically by a 3D test particle code in FORTRAN using the fourth-order Runge–Kutta method. For numerically simulation we normalize the spatial and temporal quantities with k 0 and ω0 , respectively, velocity with c and electric and magnetic field with e / m 0e cω0 . 3. Simulation results and discussion It is assumed that a pre-accelerated electron is initially injected at an angle δ with respect to the propagation direction of the laser pulse. Hence, the electron velocity is given by v0 = ˆi v0 sin δ + kˆ v0 cos δ , where v0 = 0.9c is the electron's initial velocity. It is also assumed that interaction begins when the electron reaches the origin ( x 0 = y0 = z 0 = 0 ) while the peak of the laser pulse is located at zL = −12.73 μ m. The evolution of the electron's trajectory and velocity can be obtained by simultaneous solution of Eqs. (4)–(6). The parameters of the laser pulse and the external magnetic field used in the simulation, have been presented in Table 1 and the rest parameters can be optimized by exploring dynamics of the single electron. For search of parameters that may lead to higher energy gains, in Fig. 1 we have examined the variation of the electron energy gain as a function of different parameters related to the pulsed magnetic field (according to Eq. (3)). Fig. 1a shows the variation of the electron energy gain as a function of B0 for fixed parameters t0 = 0 and τb = 424 ps. It clearly shows there are two peaks for electron energy curve and the maximum energy gain, about ε = 4.5 2
Optik - International Journal for Light and Electron Optics xxx (xxxx) xxxx
S.H. Hashemian and H. Akou
Table 1 Parameters of the laser pulse and the external magnetic field. Pulsed magnetic field
Gaussian laser pulse
Strength; B0 = 0.415 (5.6 kT) Peak of the pulse; t0 = 0
Wave length; λ 0 = 0.8 μ m Time duration; τp = 22 fs Beam waist; r0 = 40 μ m
Life time; τ b = 0.424 ns
Laser intensity; I = 2.5 × 1020 W cm −2 Initial phase; ϕ0 = 0
Fig. 1. Final electron energy gain as a function of a) the strength of external magnetic field B0 , b) the peak of the magnetic field pulse t0 , c) the life time of magnetic filed τb .
GeV, can be achieved when the normalized magnetic field B0 is set to 0.37. The relationship between ε and t0 for fixed parameters B0 = 0.415 and τb = 424 ps was also investigated, which is given in Fig. 1.b. As we indicated, t0 represents the peak position of the magnetic field pulse in time. As can be seen, the final electron energy is maximized at t0 = 148 ps and after the point, the electron energy gain is decreased by increasing t0 . In Fig. 1c, t0 = 0 and B0 = 0.415[=5.6 kT] were held constant and the duration of the pulsed external magnetic field is varied. Such value of the high intensity magnetic field with short life time (nanosecond) is currently available experimentally [37,38]. As shown in Fig. 1c, the electron energy increases exponentially with increasing the magnetic field pulse duration. This was predictable, because the increase in the duration of the pulsed magnetic field would mean increasing the time that the magnetic field affects the particle and consequently, the particle remains in the acceleration phase. By optimization, it is seen that an electron with 10.5 GeV energy can be obtained by a RHCP laser pulse with intensity 3.8 × 1021W/cm 2 corresponding to a0 = 0.61 × 10−9λ 0 [μm] I [W /cm2] = 30 in the presence of the pulsed magnetic field with B0 = 0.6, t0 = 0 and τb = 424 ps. By considering the Doppler shift effect, and by applying the external magnetic field the cyclotron resonance condition between the electron and the laser wave will be ω0 − k 0 vz = eBext (t )/ γm 0e c ≡ ωc (t )/ γ , where ωc (t ) is the non-relativistic cyclotron frequency [24]. Using dimensionless variables the resonance condition can be written as
γ (1 − βz ) Bext (t )
= 1,
(7)
Fig. 2. a) Variation of normalized external magnetic field Bext (t ) (red line) and electron energy gain ε (blue line) versus the interaction time. b) Variation of the normalized external magnetic field Bext (t ) (red line) and γ (1 − βz ) which presents the numerator of Eq. (21) (blue line) versus the interaction time. The inset shows the variation of γ (1 − βz )/ Bext (t ) versus time. c) Evolution of energy gradient dε / dt versus time. 3
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where Bext (t ) is the normalized external magnetic field. Fig. 2.a shows the pulsed external magnetic field in red line and the variation of electron energy gain in blue line as a function of the interaction time t . The maximum energy gain of 4.4 GeV is observed for the electron. To check the resonance condition, the variation of numerator and denominator of Eq. (7), versus interaction time are shown in Fig. 2.b. As can be seen, these two curves cross each other at t = 56.878 ps. This is the time which resonance condition occurs exactly as seen in the inset of Fig. 2b which shows the variation of the left side of Eq. (7). For further investigation of resonance, it is useful to consider the behaviour of the electron energy gradient over the interaction time. Using Eq. (5) the energy gradient of the electron interacting with the laser wave can given by
dε e EL . v =− . dz vz
(8)
The evolution of the acceleration gradient is shown in Fig. 2c. It is seen that exactly after t = 56.878 ps from the beginning of the interaction, the electron energy gradient reaches its maximum value. Although, in the rest of the interaction time, the auto-resonance condition does not exactly take place, in the most important part of the electron acceleration process, (0 < t < 200 ps) the fraction in the left side of Eq. (7) is not very different from 1 and the acceleration gradient is positive. In fact after this stage, the interaction practically finished. It is interesting to note that the gradient of energy is much larger than that in the conventional accelerators. After the detailed investigation of the single electron dynamics, we now consider a bunch of electrons injected into the laser acceleration configuration in the presence of the pulsed magnetic field. A random distribution of an ensemble of non-interacting electrons is considered within a volume of cylindrical shape along the z -direction. The coordinates of the electrons ( x , y, z ) are defined by generating random numbers for ρ , φ and z in cylindrical coordinate system as 0 ≤ ρ ≤ RB , 0 ≤ φ ≤ 2π and 0 ≤ z ≤ LB . Where RB and LB are respectively the radius and length of the cylinder containing the electrons. In this simulation, a normal distribution has been used to generate the random numbers for ρ , while a uniform distribution has been selected for φ and z . Therefore, the electrons in the cylinder are randomly Gaussian distributed around the z -axis. To determine the components of the initial velocity of each electron in the initial beam the following relations are used.
βx = |β0 |sin δ cos φδ , ⎧ ⎪ βy = |β0 |sin δ sin φδ , ⎨ ⎪ βz = |β0 |cos δ . ⎩ where |β0 | is the initial velocity of an electron, δ and φδ represent the axial and azimuthal angles. As can be seen in Fig. 3.a, the initial velocity of the electrons follows a normal (Gaussian) distribution with mean value β¯0 = 0.9 and standard deviation Δβ0 = 0.013. Fig. 3b shows the distribution of initial energy of the electrons. The initial mean energy of the electron beam is ε¯i = 1.17 MeV with deviation Δεi = 0.06 MeV. Such the pre-accelerated electron beam can be achieved using a short LINAC or a tabletop betatron. The space distribution of initial and output electron beam have been shown in Fig. 4a and b. The electron number density of the initial bunch is n = 3.2 × 1011cm −3 and the radial and longitudinal sizes of the cylinder are RB = 10 μm = 0.25r0 and LB = 20 μm = 25λ 0 , respectively. For further clarification of the longitudinal distribution of the output electrons, the figure is magnified along the z -direction as can be seen in Fig. 4c. According to the figures, the features of the outgoing electron beam have been presented as follows. 3.1. Hollow electron beam As can be seen in the cross-section of the electrons distribution in the x − y plane (Fig. 4b and c), the accelerated electrons leave around the z -axis toward the radial direction, therefore a hollow electron beam is formed. The cross-section forms a circular ring with a radius 30 μ m and a thickness of 20 μ m. This feature is due to the cyclotron rotation of the electrons around the axial magnetic field
Fig. 3. Percentage of total number of the injected electrons as a function of a) initial velocity β0 and b) initial energy εi . 4
Optik - International Journal for Light and Electron Optics xxx (xxxx) xxxx
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Fig. 4. a) Initial spatial distribution of electron bunch in a cylinder. b) 3D spatial distribution of the outgoing electrons and c) the magnified spatial distribution in z-direction.
which can cause the electrons to be expelled from the z -direction. Moreover, the radially ponderomotive force related to the Gaussian laser beam can push the electrons away from the axis. Fig. 5.a illustrates the percentage of total number of outgoing electrons versus radial coordinate r . It is seen that due to the ponderomotive force, the transverse plane in 0 < r < 10 μ m is empty of each particle and more particles are distributed around the r = 24 μ m. Such hollow electron beam may be used in collimation in storage rings and colliders [29] 3.2. Circulating electron beam Due to the rotation of each electron around the propagation direction caused by the applied external magnetic field, the ensemble of the particles performs a general rotational movement. This is clear in Fig. 4c. Therefore, a circulating electron beam has been produced which moves forward with high relativistic velocities. The evolution of angular momentum of electrons in z -direction is responsible for the circulation process in the transverse plane. The longitudinal component of the electron angular momentum is given by Lz = [r × P]z = γ [xβy − yβx ], which indicates all electrons rotate around the axis. The electron beam with carrying axial angular momentum can provide new capabilities for electron microscopy [30] and micro-manipulation [31,32]. 3.3. Transverse emittance Emittance is a fundamentally important parameter which can evaluate quality of a bunch of particles. That is a criterion for the particle scattering in transverse directions. The root-mean-square (RMS) emittance in the transverse direction can be obtained from the following formula [39] 1
1
ϵx = 〈 (x − 〈x 〉)2〉2 〈 (x ′ − 〈x ′〉)2〉2 ,
(9)
Fig. 5. a) Percentage of total number of the electrons in output beam versus radial coordinate. b) The spread distribution of outgoing electron beam energy. 5
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S.H. Hashemian and H. Akou
where x ′ = Px / Pz is the inclination of the electron trajectory relative to the z -axis which denotes the divergence of electrons and Px is the electron momentum. In our calculation, the initial and final transverse RMS emittances in π mm.mrad unit are respectively ϵix , y = 0.0092 and ϵ xf , y = 0.045. Thus, the ratio of the emittances of the final electron beam to the initial one is ϵ xf , y /ϵix , y = 4.89. Such values for emittance of an electron bunch are very favourable compared to other theoretical or experimental reports. 3.4. Energy distribution Another important parameter to indicate the quality of the accelerated electron bunch is its energy distribution. The simulation results show that the outgoing electrons form a high energy bullet with a length of 4λ 0 in z -direction. The mean energy gain of the electron beam is ε¯ = 3.2 GeV. Since laser wave phases experienced by each electron may be different, their energy gain is slightly different. Therefore, the energy distribution of the output beam is broadened. The broadening of the energy distribution of the output electrons has been shown in Fig. 5.b. It is seen that the energy of more electrons is about 3.3 GeV. By calculating the FWHM of the electron number in Fig. 5.b the beam energy spread is estimated to be about (Δε )1/2 = 0.5 GeV. Now we can calculate the energy emittance of the output electrons by ϵ′ = (Δε )1/2 / εp [2], where εp is the energy corresponding to the peak of energy spectrum. The energy emittance of the output electron beam is ϵ′ = 0.15, which shows that the energy divergence between the particles is very small in comparison with the other works [2,25]. This is an important issue in practical application. 3.5. Angular spectra There is a simple correlation between the scattering angle and the escape energy of the electron. The scattering angle of an output electron can be obtain by δf = arctan(P⊥/ Pz ) . Using energy conservation and relativistic factor one can also write [40]
⎡ γ2 − 1 ⎤ δf = tan−1 ⎢ − 1 ⎥. (γβz )2 ⎣ ⎦
(10)
Fig. 6.a presents the angular distribution of the outgoing electrons. As shown in the figure, the electrons are concentrated in a smallscattering angle region. In this simulation the mean scattering angle is about δ¯f = 0.66°. In Fig. 6b, we show the scattering angle of the outgoing electrons as a function of the injection angle of the electrons δi . As can be seen, while the injection angle of the electrons is in the range of 0 to 2° (Δδi = 2°), their scattering angle range is between 0.62° to 0.82° (Δδf = 0.2°). In other words, the accelerated electrons are distributed in a very small angular range. The characteristics of the initial and outgoing electron beam have been summarized in Table 2. 4. Summary and conclusions In this paper, the interaction of a single electron with a femto-second RHCP Gaussian laser pulse in the presence of a short duration axial magnetic field has been considered. The laser peak intensity is about 2.5 × 1020 W cm −2 (peak power 6.3 PW), and the peak strength of the nano-second pulsed magnetic field is about 5.6 kT . Such employed magnetic fields are currently available in spite of their great strength because of their ultra-short lifetime (τb < 1 ns) [37,38]. Due to the magnetic field, the electron rotates around the z -axis with cyclotron frequency. It is concluded that, when the cyclotron frequency coincides with the laser frequency experienced by the relativistic electron, a resonance occurs and therefore it leads to a maximal energy gradient, and a multi-GeV energy exchange between the laser wave and electron. The main result of this paper is the development of the production of a high quality electron beam with several special and novel features. The simulation presents a high-quality circulating hollow electron beam. The
Fig. 6. a) Angular distribution of outgoing electrons. b) The scattering angle of outgoing electrons (δf ) as a function of the injection angle of the electrons (δi ). 6
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Table 2 Features of the initial and output electron beam. Initial Beam
Output Beam
1.17 MeV 0.05 0.0092
3.2 GeV 0.15 0.045
Bunch diameter dB (=2RB ) Mean injection (scattering) angle δ Injection (Scattering) angle range Δδ
20 μm (≃25λ 0) 20 μ m 0.46° 2°
3.2 μm (≃4λ 0) 60 μ m 0.66° 0.2°
Electron number density [cm−3 ]
3.2 × 1011
1.8 × 1011
Mean energy ε¯ Energy emittance ϵ′[Dimensionless] Transverse Emittance ϵx , y [π mm.mrad] Bunch length LB
quality of the beam depends on its energy gain, energy spread and the particle's scattering where the last two characteristics can be expressed by energy and spatial emittances. The simulation reveals a high-energy electron beam with a mean energy of about ε¯ = 3.2 GeV and with an low energy emittance (ϵ′ = 0.15), suitable for high-precision particle physics experiments. The presence of a suitable pulsed magnetic field during the laser-particle interaction not only improves the beam features related to energy, but also amends the spatial emittance as well as the angular scattering of the electrons. The transverse emittance of the magnetized output beam is about 0.045 π mm.mrad, which compares very small with what is obtained from other theoretical reports [23,25] and conventional accelerators [41]. The simulation results show that the broadening of the electrons scattering angle relative to the broadening of the electrons injection angle is Δδf /Δδi = 0.1 which means that the outgoing electrons are distributed in a smaller angular range compared to the injection electrons. As a final word, the shape of output electron beam and its rotation add to its the practical applications. Acknowledgements The authors acknowledge the funding support of Babol Noshirvani University of Technology through Grant program No. BNUT/ 391040/98. References [1] V. Yanovsky, V. Chvykov, G. Kalinchenko, P. Rousseau, T. Planchon, T. Matsuoka, A. Maksimchuk, J. Nees, G. Cheriaux, G. Mourou, K. Krushelnick, Opt. Exp. 16 (2008) 2109 Central Laser Facility www.clf.rl.ac.uk/.. [2] P.X. Wang, Y.K. Ho, J. Pang, X.Q. Yuan, Q. Kong, N. Cao, Nucl. Instrum. Methods Phys. Res. Sect., A 482 (2002) 581. [3] D.N. Gupta, N. Kant, D.E. Kim, H. Suk, Phys. Lett. A 368 (2007) 402. [4] Y.I. Salamin, Phys. Rev. A 73 (2006) 043402. [5] P.B. Corkum, F. Krausz, Nat. Phys. 3 (2007) 381. [6] H. Zhang, Sh. Liu, H. Guo, Phys. Lett. A 367 (2007) 402. [7] A.G. Khachatryan, F.A. van Goor, K.-J. Boller, Phys. Rev. E 70 (2004) 067601. [8] X.Y. Wu, P.X. Wang, S. Kawata, Appl. Phys. Lett. 100 (2012) 221109. [9] M. Asri, H. Akou, Optik 143 (2017) 142. [10] Y.I. Salamin, Phys. Lett. A 376 (2012) 2442. [11] M. Akhyani, F. Jahangiri, A.R. Niknam, R. Massudi, J. Appl. Phys. 118 (2015) 183106. [12] F. Sohbatzadeh, S. Mirzanejhad, H. Aku, Phys. Plasmas 16 (2009) 023106. [13] F. Sohbatzadeh, S. Mirzanejhad, H. Aku, S. Ashouri, Phys. Plasmas 17 (2010) 083108. [14] F. Sohbatzadeh, H. Aku, J. Plasma Phys. 77 (2011) 39. [15] H. Akou, Phys. Plasmas 25 (2018) 063105. [16] M. Fouladi, H. Akou, J. Opt. Soc. Am. B 36 (2019) 603. [17] H. Akou, Optik 131 (2017) 446. [18] H. Liu, X.T. He, S.G. Chen, Phys. Rev. E 69 (2004) 066409. [19] D.N. Gupta, C.M. Ryu, Phys. Plasmas 12 (2005) 053103. [20] K.P. Singh, Phys. Rev. E 69 (2004) 056410. [21] D.N. Gupta, H. Suk, Phys. Plasmas 13 (2006) 013105. [22] K.P. Singh, J. Appl. Phys. 100 (2006) 044907. [23] B.J. Galow, J.-X. Li, Y.I. Salamin, Z. Harman, Ch.H. Keitel, Phys. Rev. ST. Accel. Beams 16 (2013) 081302. [24] H. Saberi, B. Maraghechi, Phys. Plasmas 22 (2015) 033115. [25] H. Akou, M. Hamedi, Phys. Plasmas 22 (2015) 103120. [26] H.S. Ghotra, N. Kant, Phys. Plasmas 23 (2016) 013101. [27] H.S. Ghotra, N. Kant, Phys. Plasmas 23 (2016) 053115. [28] D.N. Gupta, H. Suk, Appl. Phys. Lett. 91 (2007) 211101. [29] G. Stancari, A. Valishev, G. Annala, G. Kuznetsov, V. Shiltsev, D.A. Still, L.G. Vorobiev, Phys. Rev. Lett. 107 (2011) 084802. [30] S. Furhapter, A. Jesacher, S. Bernet, M. Ritsch-Marte, Opt. Lett. 30 (2005) 1953. [31] D.G. Grier, Nature 424 (2003) 810. [32] A. Mair, A. Vaziri, G. Weihs, A. Zeilinger, Nature 412 (2001) 313. [33] G. Foo, D.M. Palacios, G.A. Swartzlander, Opt. Lett. 30 (2005) 3308. [34] M.F. Anderson, C. Ryu, P. Clad, V. Natarajan, A. Vaziri, K. Helmerson, W.D. Phillips, Phys. Rev. Lett. 97 (2006) 170406. [35] K.T. McDonald, Phys. Rev. Lett. 80 (1998) 1350. [36] P. Mora, B. Quesnel, Phys. Rev. Lett. 80 (1998) 1351. [37] K.F.F. Law, M. Bailly-Grandvaux, A. Morace, S. Sakata, K. Matsuo, S. Kojima, S. Lee, X. Vaisseau, Y. Arikawa, A. Yogo, K. Kondo, Z. Zhang, C. Bellei, J.J. Santos, S. Fujioka, H. Azechi, Appl. Phys. Lett. 108 (2016) 091104.
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